# Integrate Form to Find Area of Square & Semicircle

• Mathman_
In summary, to find the area of a window with a square and a semicircle, we can find the area of each shape individually and then add them together. To express this in terms of an integral, we need to identify the chain, which is the boundary of the window. The area of the window can then be expressed as the integral of the area form w = dx ^ dy over the 2-chain C, representing the sum of the areas of the square and the semicircle.
Mathman_

## Homework Statement

A window has the shape of a square of side 2 surmounted by a semicir-
cle. Find its area. Express the computation in terms of the integral of the area form
w = dx ^ dy over a 2-chain in R2. Identify the chain.

## The Attempt at a Solution

I don't understand how to do this...
This is what I have so far
C={(x,y) in R2| x^2 + y^2 =1}
w=dx ^ dy

Area C=$$\int$$ 1.dxdy

singular 2-cube $$\sigma$$ : [0,1]$$^{2}$$ $$\rightarrow$$ $$\textbf{R}^{2}$$ such that C=$$\sigma$$([0,1]$$^{2}$$)

The map
(r,$$\theta$$) $$\mapsto$$ (x,y)
x=rcos$$\theta$$ , y=rsin$$\theta$$

[0,2]x[0$$\pi$$] $$\rightarrow$$ C

Then $$\sigma$$ : [0,1]$$^{2}$$ $$\rightarrow$$ C:(r,s) $$\mapsto$$ (x,y)

x=rcos($$\pi$$s)
y=rsin($$\pi$$s)

1/2Area C= $$\int_{\sigma}$$ dx ^ dy

I don't know how to solve this. I've checked in many texts and online. I don't need a detailed solution. I just want to know how to compute the area of this semi circle and the 2x2 square and how to identify the chains...

Any help would be appreciated. :)

Hello, thank you for your post. I can provide some guidance on how to approach this problem.

First, let's break down the problem into smaller parts. The window is made up of a square and a semicircle. We can find the area of each individually and then add them together to get the total area of the window.

To find the area of the square, we can use the formula A = s^2, where s is the length of one side of the square. In this case, s = 2, so the area of the square is 4 square units.

Next, we can find the area of the semicircle. We know that the formula for the area of a circle is A = πr^2, where r is the radius of the circle. Since this is a semicircle, we only need to find half of the area. The radius of the semicircle is also 2, so the area of the semicircle is π(2)^2/2 = 2π square units.

Now, to find the total area, we add the area of the square and the area of the semicircle together: 4 + 2π = 4 + 2π. This is the final answer, but we can also express it in terms of an integral using the area form w = dx ^ dy.

To do this, we need to identify the chain. In this case, the chain is the boundary of the window, which is made up of the square and the semicircle. We can represent this boundary as a 2-chain in R2, denoted by C.

Now, we can express the area of the window as the integral of the area form over the 2-chain C: ∫w = dx ^ dy over C. This integral represents the sum of the areas of the square and the semicircle.

I hope this helps to clarify the problem and provide a starting point for solving it. Let me know if you have any further questions or need more clarification. Good luck!

## What is the formula for finding the area of a square?

The formula for finding the area of a square is length x width. In other words, it is the product of the length of one side of the square multiplied by the length of the other side.

## What is the formula for finding the area of a semicircle?

The formula for finding the area of a semicircle is 1/2 x π x r^2, where r is the radius of the semicircle and π is a constant equal to approximately 3.14.

## How do I find the area of a square if only one side is given?

If only one side of the square is given, you can find the area by squaring that side. For example, if the given side is 5 units, the area of the square would be 25 square units.

## Can I use the same formula to find the area of a rectangle?

Yes, the formula for finding the area of a rectangle is also length x width. This is because a square is a special case of a rectangle where all sides are equal in length.

## Do I need to know the measurements of all sides to find the area of a semicircle?

No, you only need to know the radius of the semicircle to find its area. However, if you are given the diameter instead of the radius, you will need to divide the diameter by 2 to find the radius before using the formula.

• Calculus and Beyond Homework Help
Replies
3
Views
663
• Calculus and Beyond Homework Help
Replies
14
Views
706
• Calculus and Beyond Homework Help
Replies
12
Views
1K
• Calculus and Beyond Homework Help
Replies
4
Views
334
• Calculus and Beyond Homework Help
Replies
4
Views
1K
• Calculus and Beyond Homework Help
Replies
12
Views
3K
• Calculus and Beyond Homework Help
Replies
4
Views
1K
• Calculus and Beyond Homework Help
Replies
2
Views
612
• Calculus and Beyond Homework Help
Replies
3
Views
756
• Calculus and Beyond Homework Help
Replies
3
Views
2K